A within-host model on the interactions of sensitive and resistant <i>Helicobacter pylori</i> to antibiotic therapy considering immune response

Edgar Alberto Vega Noguera; Simeón Casanova Trujillo; Eduardo Ibargüen-Mondragón; Edgar Alberto Vega Noguera; Simeón Casanova Trujillo; Eduardo Ibargüen-Mondragón

doi:10.3934/mbe.2025009

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 1: 185-224. doi: 10.3934/mbe.2025009

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Research article

A within-host model on the interactions of sensitive and resistant Helicobacter pylori to antibiotic therapy considering immune response

1.
Departamento de Matemáticas y Estadística, Universidad Nacional de Colombia, Campus la Nubia, Manizales 170003, Colombia
2.
GTA ABC Dynamics, Universidad Nacional de Colombia, Campus la Nubia, Manizales 170003, Colombia
3.
Departamento de Matemáticas y Estadística, Universidad de Nariño, C.U. Torobajo, Cll 18 - Cra 50, Pasto 520002, Colombia
4.
Grupo de Investigación en Biología Matemática y Matemática Aplicada (GIBIMMA), Universidad de Nariño, C.U. Torobajo, Cll 18 - Cra 50, Pasto 520002, Colombia

Academic Editor: Libin Rong

Received: 10 October 2024 Revised: 05 December 2024 Accepted: 03 January 2025 Published: 20 January 2025

In this work, we formulated a mathematical model to describe growth, acquisition of bacterial resistance, and immune response for Helicobacter pylori (H. pylori). The qualitative analysis revealed the existence of five equilibrium solutions: (ⅰ) An infection-free state, in which the bacterial population and immune cells are suppressed, (ⅱ) an endemic state only with resistant bacteria without immune cells, (ⅲ) an endemic state only with resistant bacteria and immune cells, (ⅳ) an endemic state of bacterial coexistence without immune cells, and (ⅴ) an endemic coexistence state with immune response. The stability analysis showed that the equilibrium solutions (ⅰ) and (ⅳ) are locally asymptotically stable, whereas the equilibria (ⅱ) and (ⅲ) are unstable. We found four threshold conditions that establish the existence and stability of equilibria, which determine when the populations of sensitive H. pylori and resistant H. pylori are controlled or eliminated, or when the infection progresses only with resistant bacteria or with both bacterial populations. The numerical simulations corroborated the qualitative analysis, and provided information on the emergence of a limit cycle that breaks the stability of the coexistence equilibrium. The results revealed that the key to controlling bacterial progression is to keep bacterial growth thresholds below 1; this can be achieved by applying an appropriate combination of antibiotics and correct stimulation of the immune response. Otherwise, when bacterial growth thresholds exceed 1, the bacterial persistence scenarios mentioned above occur.
- nonlinear differential equations,
- antibiotic resistance,
- stability,
- immune system,
- H. pylori
Citation: Edgar Alberto Vega Noguera, Simeón Casanova Trujillo, Eduardo Ibargüen-Mondragón. A within-host model on the interactions of sensitive and resistant Helicobacter pylori to antibiotic therapy considering immune response[J]. Mathematical Biosciences and Engineering, 2025, 22(1): 185-224. doi: 10.3934/mbe.2025009

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Abstract

In this work, we formulated a mathematical model to describe growth, acquisition of bacterial resistance, and immune response for Helicobacter pylori (H. pylori). The qualitative analysis revealed the existence of five equilibrium solutions: (ⅰ) An infection-free state, in which the bacterial population and immune cells are suppressed, (ⅱ) an endemic state only with resistant bacteria without immune cells, (ⅲ) an endemic state only with resistant bacteria and immune cells, (ⅳ) an endemic state of bacterial coexistence without immune cells, and (ⅴ) an endemic coexistence state with immune response. The stability analysis showed that the equilibrium solutions (ⅰ) and (ⅳ) are locally asymptotically stable, whereas the equilibria (ⅱ) and (ⅲ) are unstable. We found four threshold conditions that establish the existence and stability of equilibria, which determine when the populations of sensitive H. pylori and resistant H. pylori are controlled or eliminated, or when the infection progresses only with resistant bacteria or with both bacterial populations. The numerical simulations corroborated the qualitative analysis, and provided information on the emergence of a limit cycle that breaks the stability of the coexistence equilibrium. The results revealed that the key to controlling bacterial progression is to keep bacterial growth thresholds below 1; this can be achieved by applying an appropriate combination of antibiotics and correct stimulation of the immune response. Otherwise, when bacterial growth thresholds exceed 1, the bacterial persistence scenarios mentioned above occur.

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